Why did they throw letters into it? WHY?

They're not letters! They're relationships :)

3x = 15

The x is not a letter. The x is a symbol that means "for the relationship to be true, 3 must be multiplied by a certain number so that it equals 15." You can simply rephrase that as "for the relationship to be true, a number three times smaller than 15, when multiplied by 3, equals 15", which is pretty 'duh', right?

x = 15/3  (or)  x = 5

The symbols have always ever stood for relationships. You have numbers. The relationship of numbers to each other is what we call addition, subtraction, division and multiplication. We call this "elementary math". Elementary math is the hardest math to intuitively grasp. There's no way you're going to truly understand the fundamental underlying characteristics behind division and multiplication without explaining it to you wrong until your fourth year of graduate school... However, even a 5 year old can do it "mechanically" without understanding it.

Then you have the next level of math, which is the relationship of equalities (aka "equations"). Now, rather than playing with numbers themselves, you're playing with ratios, relationships of numbers, and things like that. We usually call this algebra. Algebra is easy to "mechanically" do, but very hard to intuitively understand.

Then there's the third level of math, which is the relationship of equations to each other. Here, you start looking at properties of equations (aka "functions") as they relate to each other. What's the slope of a line at a single point? What equation is there that can have its slope described by the line I'm given? Questions like that and more make up calculus. Calculus is very easy to intuitively understand, but it can be hard to mechanically do because you need to understand it before you can get the right answer, which isn't the case with algebra.

Then you start looking at the properties of numbers themselves, the fundamental parts of math, and other interesting things like that. But, really, they're all just specific areas of the other levels above. The problem is that math is usually taught from the bottom up, aka "numbers -> algebra -> calculus" when it should be taught the other way around. We teach kids "shut up and don't bother understanding this, just do it" and then finally when we say "okay, now you can try and understand it" they go "no, shut up and shovel it down our throats". That is, if they haven't completely given up and declared an everlasting hatred against it already ;)